In the electromagnetic inverse problem, diffraction tomography using the Born approximation gives an efficient reconstruction algorithm in case of weakly scattering objects and inversion scheme by using the moment method in the reverse sequence may be used in case of strong scatterer. However, direct moment method inversion suffers from the ill-posedness in a sense that a small error or noise in the measured scattered fields causes a large error in the reconstructed profile when the total number of cells of the scatterer is larger than that of the propagating modes.
An iterative inversion technique minimizing the cost function is applied to reconstruct the high contrast lossy dielectric scatterer object up to the 1λ×λ in its size where the cost function is defined as the sum of the squared magnitude of the difference fields between the measured and calculated scattered fields from the assumed set of dielectric profile.
The number of propagating modes is proportional to the radius of the scattering object, while the total number of cells is proportional to the square of radius. As the size of the scatterer increases, the total number of cells may be equal to or less than the propagating modes and the additional incident plane waves are employed for better regularization in the presence of the noise, where the total propagating modes considering different incident waves exceeds the total number of cells.
Iterative inversion using the Levenberg-Marquardt algorithm, which is one of the steepest descent Newton-like algorithms, gives rather limited success in the reconstruction of the large dielectric object, because it converges to the dielectric profile which corresponds to the local minimum of the cost function. One may study the nature of this local minima of the cost function by calculating the cost function as a function of the initial values of dielectric profile for a scattering of electromagnetic waves by a homogeneous dielectric circular cylinder.
One may show numerically that there exists many local minima even if one uses wide frequency band signals although its depth may be shallowed but it does not disappear. One may add noise to the scattered field and show numerically that the difference between the depth of the global minimum and local minima decreases and sometimes to the comparable depth of the global minimum.
Another stochastic algorithm, simulated annealing algorithm is introduced in order to force this iterative scheme to reach the global minimum all the time. A numerical example of the dielectric object of 8 × 4 cells with a quarter wavelength size square cells and of maximum relative dielectric constant of 8. Although this algorithm finds the global minimum, it takes a long computer time and a new hybrid algorithm utilizing both the Levenberg-Marquardt and the simulated annealing algorithms is proposed. One may reach via LM algorithm to one of the local minimum and SA algorithm is used to get out of this local minimum of the cost function and again use LM and SA algorithms until it reaches the global minimum. Numerical examples show that this hybrid algorithm is about 5 times faster than the SA algorithm.
The numerical efficiencies and accuracies of these iterative algorithms depend upon the various physical parameters such as the size of the cell, number of propagating modes and incident waves. These parameters are tried to be optimized on the basis of physical meanings and numerical experiences.